Shariat et al previously investigated the possibility of predicting, from preoperative biomarkers and clinical data, which of any pair of patients would suffer recurrence of prostate cancer first. We wished to establish the extent to which predictions of time of relapse from such a model could be improved upon using Bayesian methodology. The same dataset was reanalysed using a Bayesian skew-Student mixture model. Predictions were made of which of any pair of patients would relapse first and of the time of relapse. The benefit of using these biomarkers relative to predictions made without them, was measured by the apparent Shannon information, using as prior a simple exponential attrition model of relapse time independent of input variables. Using half the dataset for training and the other half for testing, predictions of relapse time from the strict Cox model gave $-\infty$ nepers of apparent Shannon information, (it predicts that relapse can only occur at times when patients in the training set relapsed). Deliberately smoothed predictions from the Cox model gave -0.001 (-0.131 to +0.120) nepers, while the Bayesian model gave +0.109 (+0.021 to +0.192) nepers (mean, 2.5 to 97.5 centiles), being positive with posterior probability 0.993 and beating the blurred Cox model with posterior probability 0.927. These predictions from the Bayesian model thus outperform those of the Cox model, but the overall yield of predictive information leaves scope for improvement of the range of biomarkers in use. The Bayesian model presented here is the first such model for prostate cancer to consider the variation of relapse hazard with biomarker concentrations to be smooth, as is intuitive. It is also the first model to be shown to provide more apparent Shannon information than the Cox model and the first to be shown to provide positive apparent information relative to an exponential prior.

翻译：Shariat等人先前研究了通过术前生物标志物和临床数据预测任意一对患者中哪一位会先发生前列腺癌复发的可能性。我们旨在确定使用贝叶斯方法在多大程度上能改善此类模型对复发时间的预测效果。采用贝叶斯偏斜学生混合模型对同一数据集重新进行分析，预测了任意一对患者中谁将先复发以及具体的复发时间。以与输入变量无关的简单指数衰减复发时间模型作为先验，通过表观香农信息衡量使用这些生物标志物相较于不使用时的获益程度。使用一半数据集进行训练、另一半进行测试时，严格Cox模型对复发时间的预测产生了-∞奈特的表观香农信息（该模型预测复发仅可能发生在训练集患者复发的对应时间点）。经刻意平滑处理的Cox模型预测结果为-0.001（-0.131至+0.120）奈特，而贝叶斯模型预测结果为+0.109（+0.021至+0.192）奈特（均值及2.5至97.5百分位数），其后验概率为0.993时呈正值，并以0.927的后验概率优于模糊Cox模型。因此，贝叶斯模型的预测性能优于Cox模型，但预测信息的整体产出仍为现有生物标志物范围的改进留有余地。本文提出的贝叶斯模型是首个考虑复发风险随生物标志物浓度平滑变化（符合直觉）的前列腺癌模型，也是首个被证明比Cox模型提供更多表观香农信息、且相对于指数先验提供正表观信息的模型。